MathDB
Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2012 Switzerland - Final Round
2012 Switzerland - Final Round
Part of
Switzerland - Final Round
Subcontests
(7)
9
1
Hide problems
1+ab+bc+ca >= min { (a + b)^2 / ab } for a,b,c>0 with abc = 1
Let
a
,
b
,
c
>
0
a, b, c > 0
a
,
b
,
c
>
0
be real numbers with
a
b
c
=
1
abc = 1
ab
c
=
1
. Show
1
+
a
b
+
b
c
+
c
a
≥
min
{
(
a
+
b
)
2
a
b
,
(
b
+
c
)
2
b
c
,
(
c
+
a
)
2
c
a
}
.
1 + ab + bc + ca \ge \min \left\{ \frac{(a + b)^2}{ab} , \frac{(b+c)^2}{bc} , \frac{(c + a)^2}{ca}\right\}.
1
+
ab
+
b
c
+
c
a
≥
min
{
ab
(
a
+
b
)
2
,
b
c
(
b
+
c
)
2
,
c
a
(
c
+
a
)
2
}
.
When does equality holds?
8
1
Hide problems
compare no of ways for 2 paths along surface of cube
Consider a cube and two of its vertices
A
A
A
and
B
B
B
, which are the endpoints of a face diagonal. A path is a sequence of cube angles, each step of one angle along a cube edge is walked to one of the three adjacent angles. Let
a
a
a
be the number of paths of length
2012
2012
2012
that starts at point
A
A
A
and ends at
A
A
A
and let b be the number of ways of length
2012
2012
2012
that starts in
A
A
A
and ends in
B
B
B
. Decide which of the two numbers
a
a
a
and
b
b
b
is the larger.
7
1
Hide problems
sum of all factors of n3k +2 is divisible by 3
Let
n
n
n
and
k
k
k
be natural numbers such that
n
=
3
k
+
2
n = 3k +2
n
=
3
k
+
2
. Show that the sum of all factors of
n
n
n
is divisible by
3
3
3
.
1
1
Hide problems
2012 chameleons sitting at a round table, red or green
There are 2012 chameleons sitting at a round table. At the beginning each has the color red or green. After every full minute, each chamaleon, which has two neighbors of the same color, changes its color from red to green or from green to red. All others keep their color. Show that after
2012
2012
2012
minutes there are at least
2
2
2
chameleons that have the same often changed color.[hide=original wording]Es sitzen 2012 Chamaleons an einem runden Tisch. Am Anfang besitzt jedes die Farbe rot oder grun. Nach jeder vollen Minute wechselt jedes Cham aleon, welches zwei gleichfarbige Nachbarn hat, seine Farbe von rot zu grun respektive von gr un zu rot. Alle anderen behalten ihre Farbe. Zeige, dass es nach 2012 Minuten mindestens 2 Chamaleons gibt, welche gleich oft die Farbe gewechselt haben.
5
1
Hide problems
$-element subsets of {1, 2, . . . , n}
Let n be a natural number. Let
A
1
,
A
2
,
.
.
.
,
A
k
A_1, A_2, . . . , A_k
A
1
,
A
2
,
...
,
A
k
be distinct
3
3
3
-element subsets of
{
1
,
2
,
.
.
.
,
n
}
\{1, 2, . . . , n\}
{
1
,
2
,
...
,
n
}
such that
∣
A
i
∩
A
j
∣
≠
1
|A_i \cap A_j | \ne 1
∣
A
i
∩
A
j
∣
=
1
for all
1
≤
i
,
j
≤
k
1 \le i, j \le k
1
≤
i
,
j
≤
k
. Determine all
n
n
n
for which there are
n
n
n
such that these subsets exist.[hide=original wording of last sentence]Bestimme alle n, fur die es n solche Teilmengen gibt.
4
1
Hide problems
no infinte sequance of primes with p_{k+1} = 2p_k - 1 or p_{k+1} = 2p_k + 1
Show that there is no infinite sequence of primes
p
1
,
p
2
,
p
3
,
.
.
.
p_1, p_2, p_3, . . .
p
1
,
p
2
,
p
3
,
...
there any for each
k
k
k
:
p
k
+
1
=
2
p
k
−
1
p_{k+1} = 2p_k - 1
p
k
+
1
=
2
p
k
−
1
or
p
k
+
1
=
2
p
k
+
1
p_{k+1} = 2p_k + 1
p
k
+
1
=
2
p
k
+
1
is fulfilled. Note that not the same formula for every
k
k
k
.
2
1
Hide problems
f (f(x) + 2f(y)) = f(2x) + 8y + 6
Determine all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that for all
x
,
y
∈
R
x, y\in R
x
,
y
∈
R
holds
f
(
f
(
x
)
+
2
f
(
y
)
)
=
f
(
2
x
)
+
8
y
+
6.
f (f(x) + 2f(y)) = f(2x) + 8y + 6.
f
(
f
(
x
)
+
2
f
(
y
))
=
f
(
2
x
)
+
8
y
+
6.